Area ≈ h/2 × (y₀ + 2y₁ + 2y₂ + ... + 2yₙ₋₁ + yₙ)
Or in words: Area ≈ (h/2) × (first y + last y + 2 × all middle y-values)
Where:
- h = equal width of each strip
- y₀, y₁, ..., yₙ = y-values at equally spaced x-values (strip boundaries)
- The first and last y-values appear once; all intermediate y-values appear twice
Example: Estimate the area under a curve using 4 strips of equal width h = 2, with y-values: y₀ = 3, y₁ = 5, y₂ = 8, y₃ = 7, y₄ = 4.
Area ≈ (2/2) × (3 + 2(5) + 2(8) + 2(7) + 4)
Area ≈ 1 × (3 + 10 + 16 + 14 + 4)
Area ≈ 1 × 47 = 47 square units
This deep dive covers The Trapezium Rule Formula within Area Under Curves for GCSE Mathematics. Revise Area Under Curves in Graphs for GCSE Mathematics with 9 exam-style questions and 10 flashcards. This topic appears regularly enough that it should still be part of a steady revision cycle. It is section 4 of 11 in this topic. Use this deep dive to connect the idea to the wider topic before moving on to questions and flashcards.
Practice questions for Area Under Curves
On a velocity-time graph, what does the area under the curve represent?
Explain whether the trapezium rule gives an overestimate or underestimate for the area under the curve y = x² between x = 0 and x = 3. Justify your answer.